Poorly monitored river flows in many regions of the world
have been hindering our ability to accurately estimate global water budgets
as well as the variability of the global water cycle. In situ gauging sites,
as well as a number of satellite-based systems, make observations of river
discharge throughout the globe; however, these observations are often sparse
due to, for example, the sampling frequencies of sensors or a lack of
reporting. Recently, efforts have been made to develop methods to integrate
these discrete observations to gain a better understanding of the underlying
processes. This paper presents an application of a fixed interval Kalman
smoother-based model, called inverse streamflow routing (ISR), to generate
spatially and temporally continuous river discharge fields from discrete
observations. The method propagates the observed information across all
reachable parts of the river network (up/downstream from gauging point) and
all reachable times (before/after observation time) using a two-sweep
procedure that first propagates information backward in time to the furthest
upstream locations (inverse routing) and then propagates it forward in time
to the furthest downstream locations (forward routing). The ISR methodology
advances prediction of streamflow in ungauged basins by accounting for a
physical representation of the river system that is not generally handled
explicitly in more-commonly applied statistically based models. The key
advantages of this approach are that it (1) maintains all the physical
consistencies embodied by a diffusive wave routing model (flow confluence
relationships on the river network and the resulting mass balance, wave
velocity, and diffusivity), (2) updates the lateral influx (runoff) at the
pixel level (furthest upstream) to guarantee exhaustive propagation of
observed information, and (3) works both with a first guess of initial river
discharge conditions from a routing model (assimilation) and without a first
guess (pure interpolation of observations). Two sets of experiments are
carried out under idealized conditions and under real-world conditions
provided by United States Geological Survey (USGS) observations. Results show
that the method can effectively reproduce the spatial and temporal dynamics
of river discharge in each of the experiments presented. The performance is
driven by the density of the gauge network as well as the quality of the data
being assimilated. We find that when assimilating the actual USGS
observations, the performance decreases relative to our idealized scenario;
however, we are still able to produce an improved discharge product at each
validation site. With further testing, as well as global application, ISR may
prove to be a useful method for extending our current network of global river
discharge observations.
Introduction
In the application of water resources for human use, as well as the
monitoring and prediction of global hydrologic hazards, such as floods and
droughts, a comprehensive understanding of globally distributed runoff and
river discharge is extremely important. In many regions of the world, river
flows are poorly covered by in situ observations, and the collection of the
available observations for consumption by global end-users has proven to be a
difficult challenge, as evidenced by the available records from the Global
Runoff Data Center (GRDC; Fekete et al., 2012). Streamflow records are
typically most complete in relatively developed and populous parts of the
world; however, streamflow data in many regions are often considered
proprietary, resulting in, among other issues, difficult problems in the
management of water resources in transboundary rivers (see, e.g., Biancamaria
et al., 2011; Pavelsky et al., 2014).
Besides the usefulness of global near-real-time river discharge data for
water management, there is a great need for observations of river discharge
data to further our understanding of the global water cycle and its
representation by reanalysis and climate models. While other observational
sources for the terrestrial water budget have become more readily available
from satellite remote sensing, the lack of comprehensive river discharge
observations has resulted in a key flux (runoff) in climate models and
large-scale land surface models being poorly constrained by observations over
much of the global land surface (Sahoo et al., 2011). Furthermore, the amount
of water stored at the land surface and its time–space variability are poorly known. To better serve the global
hydrologic community, there is then a need for methods which can make further
use of the currently available global discharge data sources.
These data sources can be divided into the following sets: (1) observations
based on in situ measurements (gauges); (2) estimates based on remotely
sensed observations (e.g., satellite altimetry, synthetic aperture radar); and
(3) estimates based on land surface models (LSMs) and routing models (Pan and
Wood, 2006). Traditionally, the data provided by sets 1 and 2 can be thought
of as point observations of river discharge along a river network, whereas
set 3 can provide us with a spatially distributed representation of discharge
throughout our basins of interest, derived from modeled runoff fields. These
two variables can be connected by the process of streamflow routing, where
the spatially distributed runoff generated at the land surface flows over the
hillslope and through a river network to become streamflow in the river
channels. From this process we can then say that the streamflow (as measured
at specific points in space and time) is the integrated response to the
runoff through a subset of time and space. Due to this process, all studies
using streamflow as representative of basin runoff are limited to
applications where the temporal differences can be ignored or accounted for
(e.g., for long-term studies the aggregation of the runoff data allows us to
ignore the temporal differences; Sahoo et al., 2011; Sheffield et al., 2009;
Pan et al., 2012). There is then a need for new methods that are able to
derive spatially and temporally continuous records of runoff and river
discharge from the available data sources.
The goal of this study is to present an application of a methodology by which
we can integrate and use the point-scale observations of river discharge to
derive a product that is spatially and temporally continuous. One possible
method is the combination of point observations with spatially distributed
model estimates through assimilation. Due to the integrated nature of the
streamflow generation process, any discharge assimilation must be able to
propagate information throughout the range of influence in a basin for any
given gauge. Additionally, we must be able to assimilate all available
observations in a basin simultaneously in time and space, to resolve
conflicts due to observational errors. There have been a number of recent
studies investigating the potential for such discharge assimilation using a
wide variety of methods, (e.g., Andreadis et al., 2007; Biancamaria et al.,
2011; Paiva et al., 2013; Pan and Wood, 2013). While these methods are often
robust and comprehensive, the large computational burden, in particular for
those using ensemble Kalman filters such as that used by Andreadis et
al. (2007), limits their potential for rapid global application. In addition,
many of these methods simply adjust discharge in a forward sense and do not
fully account for the upstream spatial and temporal correlations of the
streamflow generation process. Alternative methods have focused on the use of
kriging-based statistical techniques to derive spatially distributed
estimates of river discharge, (e.g., Blöschl et al., 2013; Paiva et al.,
2015; Yoon et al., 2013). These statistical methods often show good agreement
for the reconstructed discharge with little computational cost, but are
highly dependent on the formulation of the covariance matrix for each river
system. Here we propose the use of an assimilation and interpolation scheme
for creating spatially complete and temporally continuous river discharge
records from point observations based on the inverse streamflow routing (ISR)
model, which was previously used by Pan and Wood (2013) for the generation of
spatially distributed runoff fields to be used in land surface modeling
applications, such as the calibration of model parameters. The new approach
maintains the important structure of the streamflow generation process with a
relatively low computational burden and guarantees an exhaustive propagation
of observed information to all reachable locations across the river network
and reachable times. This can be more effective compared to the assimilation
and interpolation methods discussed previously, which perform assimilation by
adjusting the state variables (e.g., water height/volume, flow rate) and
propagate the observed information much less exhaustively. Additionally,
compared with the statistical methods commonly used in the Prediction in
Ungauged Basins (PUB) initiatives (Blöschl et al., 2013), one of the key
differentiators is that the ISR method accounts for a physical representation
of the river system (in the form of a river routing model, albeit a simple
one), which should provide better physical consistency than with a purely
statistical method.
Methods
In short, the proposed method tries to propagate the observed discharge
information across all reachable parts of the river network (up/downstream
from gauging point) and all reachable times (before/after observation time)
using a two-sweep procedure that first propagates information backward in
time to the furthest possible upstream (inverse routing) and then propagates
it forward in time to the furthest possible downstream (forward routing).
Figure 1 provides a detailed illustration of the proposed scheme. The first
sweep of the procedure, known as ISR, developed by Pan and Wood (2013) to
generate spatially distributed runoff fields, plays a key role here (left
side of Fig. 1). The ISR helps to guarantee an exhaustive propagation of
observed information by updating the boundary influx (runoff) at the pixel
level (the furthest possible upstream) throughout the entire spatial and
temporal domains. The second sweep simply re-runs the same routing model
forward using the runoff fields derived from the first sweep to reconstruct
continuous discharge values everywhere (right side of Fig. 1). Since ISR does
not require an initial guess of discharge from the routing model (Pan and
Wood, 2013), the proposed method works for both data assimilation (if an
initial guess exists) and pure interpolation of observations (without an
initial guess). Given that the discharge records are ultimately created by a
routing model, this approach preserves all the physical consistencies
embodied by the chosen routing model and its parameters such as the flow
confluence relationship on the river network and the resulting mass balance,
wave velocity, and diffusivity (if a diffusive wave routing model is used).
Such a strong physical consistency can hardly be implemented by methods based
on statistical correlations between different gauging points or different
state variables in the routing model as, for example, in the river kriging
method (Paiva et al., 2013). When used as an interpolator, the proposed
method can also exactly reproduce the input observations at gauging
locations/times (Pan and Wood, 2013). As such, this method can be seen as
performing both data assimilation and interpolation. While it is true that
the experiments with and without an initial guess can be seen as “data
assimilation”, such a distinction is very important for satellite remote
sensing applications such as the future SWOT mission because estimation with
no initial guess will enable the use of “satellite-only” products, which is
equivalent to performing “interpolation”, instead of “data assimilation”
(i.e., satellite–model combined products). The mathematical formulation of
this method is described below.
Two-sweep procedure for spatiotemporal assimilation–interpolation of
discharge records. The first sweep (c) propagates observed information collected at
gauging points upstream and backward in time following the inverse streamflow
routing method developed in Pan and Wood (2013) and derives continuous runoff
fields (lateral influx at furthest possible upstream). The second
sweep (d) propagates information downstream and forward in time
(regular routing) to create continuous discharge values everywhere. The
stacked spatial maps at the top illustrate how the observed information at a
single point in time–space propagates backward in time–space (a) and
how discharge is reconstructed from the integrated runoff (b).
Routing model formulation
The basic routing model selected for this work is the University of
Washington (UW) routing model (Lohmann et al., 1996; Nijssen et al., 2001),
which provides a simple linear routing scheme that is commonly coupled with
LSMs. This model routes runoff through two processes. The first of these is
the drainage of the runoff water within a grid cell to the outlet of the
grid cell as governed by a known unit hydrograph function (UHF). This is
given by Eq. (1) below, where u(t) is the
UHF, r(t) is the pixel runoff, and o(t) is the pixel outflow.
ot=∫0trt-τuτdτ
The second process then governs the travel of water in channels between
pixels through the one-dimensional diffusive wave equation. This is given by
Eq. (2), where q is the streamflow
generated by the pixel outflow at a distance x downstream, C is the
channel wave velocity, and D is diffusivity.
dqdt=Dδ2qδx2-Cδqδx
This model is linear as long as the parameters C and D are assumed not
to be a function of the streamflow, i.e., retention effects such as lakes
and reservoirs as well as human management are not considered, and thus it
is a good candidate for our inversion. These two stages of the routing
process are then solved together using the form presented by Eq. (3) below, where i(x,t) is the impulse response
function as defined by Eq. (4).
3q(xt)=∫0trt-τut-τix,tdτ4ix,t=x2tπtDexp-(Ct-x)24Dt
By integrating Eq. (3) for all upstream pixels, denoted as allg, for any given gauge g at discretized time steps, we can determine
the streamflow at any gauge location, Q(g,t) as shown in Eq. (5).
Qg,t=∑all(g)q(x,t)
This formulation serves as the basic routing model for inversion and for the
final reconstruction of discharge from the inverted runoff fields.
Inverse streamflow routing model
Using the routing model presented above, the fixed interval Kalman smoother
can now be established for the inversion process following Pan and Wood (2013). First, the routing model must be written in a linear state space
form as a function of input states as seen in Eq. (6).
yt=H0xt+H1xt-1+⋯+Hkxt-k+ϵt
In this form, yt is a vector of the discharges at a number of gauges in
the basin and xt is a matrix of the runoff for all cells at time t.
Because of the integration to determine flow at each gauge, the model
requires runoff information up to a lag time of k+1 steps, which is the
travel time of the basin. Finally, Ht represents the measurement
operator matrix, whose elements represent the amount of runoff that one
specific cell will contribute to each gauge at a given time. These values
are calculated from the impulse response function. As a result of the
integration described above, there is a need for the solution of this
inverse problem for multiple time steps at once, which gives rise to the
fixed interval smoothing component of this inversion. Through a time
augmentation, the model can ultimately be written in the Kalman filter form
as shown in Eq. (7) below (Pan and Wood,
2010).
x^t′′=x^t′+Kt(yt′-H′x^t′-L′x^t-k′)
Here, x^t′ is the initial guess of the time augmented runoff
fields, yt′ is the time augmented streamflow measurements, H′ and
L′ are time augmented measurement operators, Kt is the Kalman gain as
given by Eq. (8), and x^t′′ is the updated estimate of the runoff
fields.
Kt=PtH′T(H′PtH′T+Rt)-1
The Kalman gain represents a weighting of the update to the runoff fields
and is controlled by Pt, which represents the error covariance
matrix of the initial forecast for the runoff, and Rt, which is the
error covariance matrix of the gauge measurements. For this study, we
perform a set of idealized experiments in which we set Rt equal to 0,
such that the inversion process provides a maximum correction to the initial
runoff guess. The error covariance (Pt) is defined as a diagonal matrix
of the long-term mean runoff error variance. In practical applications this
error term will be derived from the error utilized in the particular form of
the discharge observations. It should be noted that this method can function
without an estimate of the initial runoff conditions (a null field) and
thus, it also works for streamflow interpolation in which river discharge is
reconstructed purely from observations. With the first ISR completed through
ISR, the second sweep of flow reconstruction is done by running the same
routing model in a forward sense with the new runoff influxes.
Experimental design and study area
For this study we perform two sets of streamflow interpolation experiments
over the Ohio River basin. The Ohio River basin, along with the Tennessee
River in the southern part of the basin, is a large basin covering an area of
approximately 490 000 km2. This basin contains a wide variety of river
sizes that drain a mix of developed, undeveloped, and agricultural areas, all
of which are monitored by the United States Geological Survey (USGS) with a
dense network of gauges. This monitoring network makes the basin a good
candidate for these streamflow interpolation experiments, as we will be able
to use the extensive USGS observations as another set of data inputs.
The first of these experiments performs the inversion using synthetically
created streamflow values as a proof of concept for the method. The goal of
this experiment is to see if the streamflow interpolation method can generate
the true discharge, given varying levels of information about the prior
runoff conditions in the basin. The second experiment is the same as the
first, except that the synthetic gauge data are replaced with actual USGS
gauge data. In this experiment, the performance of the ISR method is
evaluated under “real-world” conditions given that the routing model does
not account for the effects of human management and will produce streamflows
that are likely different from the observed streamflows. Flow charts of these
two experiment sets can be seen in Fig. 2. Each of these experiments were run
for the entirety of 2009. This period was selected because the daily
discharge characteristics were representative of the climatology, with some
individual high flow events but no dramatic/extended droughts or floods. Such
a choice can minimize the impact of many compounding factors like the
deficiencies of simple diffusive wave routing under extreme conditions (high
flow and low flow). In addition, experiments and observations over a
“typical” period can be more generalizable. Based on the previous work of
Pan and Wood (2013), the wave velocity parameter and the smoothing window for
the Ohio River basin were set at 1.4 m s-1
and 70 d, respectively.
(a) Overall process flow diagram for the synthetic experiments with
the ISR model, where p is the input precipitation for the VIC LSM
distributed over the study domain and study period, r is the runoff fields
distributed over the same space, and q is discharge at discrete points
during the study period. The superscripts “Init” and “Syn” represent the
initial guess and the synthetic truth, respectively, while the “Inv”
superscript indicates the products resulting from the model. (b) Overall
process flow diagram for the ISR model, substituting actual USGS discharge
observations for the synthetic truth of the previous experiments.
Data
In each of the experiments, the NLDAS 0.125∘ meteorological data set
(Cosgrove et al., 2003) is used to force the Variable Infiltration Capacity
(VIC) LSM (Liang et al., 1994, 1996) to produce runoff fields that are
considered the “true” runoff. The NLDAS precipitation forcings were chosen
for this experiment as they combine hourly radar analyses and daily gauge
observations and are considered to provide a comprehensive and reliable set
of forcings over the United States (Pan et al., 2010). This NLDAS-derived
runoff is then used with the routing model described previously to generate
synthetic streamflow values at set evaluation sites (“pseudo gauges”) for
the study period. These 75 sites are the routing model grid cells in which an
actual USGS gauge is located. 25 of these gauge sites are designated as
validation sites and the remaining 50 sites provide river discharge time
series to be assimilated in the ISR model. The selection of these gauge sites
was based solely on finding gauges within the basin that had relatively
complete discharge records (> 95 % days available) for the
year 2009 and the distribution of validation sites was random. The use of
these USGS gauge-based sites for validation of the synthetic model results
also allows for later experiments and comparisons with the actual USGS
observations. The distribution of these pseudo gauge stations as well as a
representation of the routing model river basin can be seen in Fig. 3.
The Ohio River basin modeled at 0.125∘ resolution and the
distribution of the 75 USGS gauge sites used in the creation of pseudo gauges
for assimilation. Blue dots represent those gauges used in the assimilation
and interpolation while the red dots represent those gauges which will be
reconstructed for evaluation. The background shading indicates the travel
time from each grid cell to the outlet of the basin.
The generated synthetic streamflows are considered the true observations and
are used in the streamflow interpolation process to correct an initial
estimate of river discharge (derived from an initial estimate of daily runoff
that is also routed using the Lohmann routing model). To investigate the
impact of this initial runoff estimate, we perform each synthetic experiment
with three daily initial conditions. These are (1) a long-term mean value of
runoff applied over the entire basin (same value in every grid cell for every
day in the study period), (2) a daily climatology of runoff at each grid
cell, derived from the NLDAS forced VIC LSM, and (3) daily runoff values
derived from the VIC LSM forced with the real-time TRMM Multi-Satellite
Precipitation Analysis (TMPA) version 3B42RT (Huffman et al., 2007)
precipitation product. The TMPA product was selected for this experiment as
it is globally available between 60∘ N and 60∘ S at a 3 h
temporal and 0.25∘ spatial resolution. The product was interpolated
to 0.125∘ to force the VIC simulations (Pan et al., 2010). While this
product is not as accurate as the ground observation-based NLDAS product, it
is globally available and can be used along with the VIC LSM to provide us
with a realistic initial forecast of runoff even when ground observations do
not exist (Pan et al., 2010). The results of these three purely synthetic
experiments and three USGS observation-based experiments are presented in
Sect. 3.
Results
Following the above methodology, six discharge interpolation (reconstruction)
experiments were performed. To evaluate the performance of the interpolation
in each of these experiments we compute the Nash–Sutcliffe efficiency (NSE)
at each of the 25 pseudo gauges designated for validation in Fig. 3. The NSE
is a measure of model performance and is defined in equation (9), where
Qo is the mean of observed discharges, Qmt is
modeled discharge at time t, and Qot is observed discharge
at time t (Nash and Sutcliffe, 1970).
NSE=∑t=1TQmt-Qot2∑t=1TQot-Qo‾2
The NSE may range from -∞ to 1, with an efficiency of 1 meaning that
there is a perfect match between the modeled discharge and the observations
(or the synthetic truth). An efficiency of 0 indicates that the model is just
as accurate as the mean of the observations, and a value less than 0
indicates that the mean would be a better predictor than the model.
In addition, to more comprehensively evaluate the performance of these
discharge reconstructions, the Kling–Gupta efficiency (KGE) and its
component statistics including correlation coefficient (r), bias ratio
(β), and relative variability (α) are calculated for each
experiment, following Eqs. (10) to (13). In these equations, Qm
and Qo denote modeled and observed discharge, respectively, while
σQm and σQo denote standard deviation
of the modeled and observed discharge, respectively. KGE measures the
Euclidean distance between a point and the optimal point that has r=1,
β=1, and α=1, producing a desired KGE of 1 (Gupta et al.,
2009; Kling et al., 2012). As a result of this, KGE is an integrated skill
metric through which one can jointly consider the modeled time series
co-variability with the observations (or synthetic truth), the model bias,
and the model standard error (Gupta et al., 2009).
10KGE=1-(r-1)2+(β-1)2+(α-1)211r=cov(Qm,Qo)σQm⋅σQo12β=Qm‾/Qo‾13α=σQm/Qm‾σQo/Qo‾
Synthetic discharge interpolation
For the first set of experiments we follow the procedure outlined by Fig. 2a.
An example of the discharge interpolation can be seen in Fig. 4, where the
time series of discharge are shown for 2 of the 25 validation gauges. The
runoff initial conditions for this set of reconstructions was the
climatological daily runoff. Figure 4a, which represents a downstream gauge
with a large upstream area, shows the performance of the ISR method. We find
that the NSE increased from 0.527 to 0.995, indicating a large increase in
the model performance through assimilation. By examining the overall time
series, it is evident that the assimilation was able to correct for a
majority of the conditions imposed by the initial guess of runoff. For
example, between days 50 and 100 the initial guess had significantly higher
flows compared to the synthetic truth, where these high flows centered around
day 50, and the assimilation was able to reconstruct this quite well.
Figure 4b shows the same results for an upstream gauge with a smaller
contributing area, where we observe an increase in NSE from 0.049 to 0.986
after the discharge reconstruction. Similar to the previous example, the ISR
methodology was effective in reconstructing spatially and temporally
continuous discharge records. Despite this, we find that for some gauges with
the smallest upstream areas, which potentially contain less assimilated
gauges than others, the reconstructions will occasionally miss the temporal
dynamics of the synthetic truth, such as between days 150 and 200 in Fig. 4b.
Reconstructed discharge time series during 2009 for 2 of the 25
evaluation sites when the ISR model was run using the climatological initial
guess of runoff conditions. The blue line represents the synthetic truth
discharge used in the inverse routing, the black line illustrates the
discharge derived from our initial guess of runoff, and the red line
illustrates the reconstructed discharge. NSE values are given for the initial
guess and the reconstruction in relation to the synthetic truth.
Figure 5 illustrates the evaluation of the KGE values across all of the
validation sites for each of the three initial runoff conditions. These plots
of the distribution of KGE values in the validation gauge set for the initial
guess and the reconstructed discharge illustrate the performance improvement
from the streamflow interpolation method. Across all of the initial
conditions there is an increase in performance for many of the gauges, with a
shift in the KGE values towards 1. In particular, we find that the null
initial guess of runoff performs the best (Fig. 5d). This is likely because
we are not imposing any temporal or spatial dynamics on the runoff, just a
mean value, which allows the interpolation to adequately reconstruct the
temporal dynamics of the synthetic truth. In particular, the initial
discharges are not very biased (Fig. 5g) but have very poor correlation and
variability (Fig. 5j, m). The improvement in both correlation and variability
illustrates the ability of the ISR methodology to significantly improve upon
the initial guess. In contrast to this, we found that the experiment with
initial conditions based on the TMPA-observed precipitation performed the
worst, as there were often differences in when events such as high flows
started or the magnitude of these events, which the assimilation was not able
to completely correct for. Despite these differences, we find that the ISR
method is able to create discharge records across all initial conditions,
with a noticeable increase in performance in each case due to the improvement
in correlation and variability.
Distributions of NSE values (a, b, c), KGE
values (d, e, f), and its component statistics: bias ratio β(g, h, i), correlation coefficient r(j, k, l), and
relative variability α(m, n, o), for the three synthetic
experiments with varied initial conditions of runoff (shown in three
columns). These daily initial conditions are (1) null (uniform mean runoff
over the entire basin), (2) climatology (average daily runoff over the entire
period from NLDAS), and (3) TMPA (runoff derived from TMPA precipitation and
VIC LSM). In each plot, the red bars illustrate the distribution of the
statistic values for discharge generated from the initial guess of runoff and
the blue bars indicate the same distribution after reconstruction with the
inverse routing method.
To further illustrate the impact of upstream area and gauge density on the
performance of the interpolation, we plot the upstream area of each
validation gauge versus the NSE for reconstructed discharge in Fig. 6. For
each experiment the same pattern is observed, with a wide variety of NSE
values for gauges with upstream areas of less than 104 km2, while
basins larger than this have NSE values consistently between 0.9 and 1. These
larger sub-basins incorporate the information of other assimilated upstream
gauges, allowing for more accurate
reconstruction of discharge. In addition to this, the integrative nature of
the routing and smoothing procedure dampens many of the short-term high flow
events, allowing the larger sub-basins to exhibit consistently better
performance given reliable upstream observations.
Distribution of NSE values for each of the evaluation sites versus
the size of the upstream area for each gauge. The ordering and experiment
names are the same as those in Fig. 5.
USGS gauge interpolation
In addition to these purely synthetic experiments, we evaluated the
performance of the streamflow interpolations under real-world conditions by
substituting daily observed USGS river discharge values for the synthetic
truth used previously. Here we present the results of these three USGS-based
experiments, varying the initial runoff conditions in the same manner as the
previous experiments. Figure 7 illustrates the performance of the ISR model
for discharge reconstruction when assimilating these in situ river discharge
observations. Again, we find that the method works well for the two
evaluation gauges presented, with the larger basin (Fig. 7a) improving the
NSE from 0.166 to 0.862 and the smaller basin (Fig. 7b) improving from
-0.061 to 0.942. Comparing these results to those from the purely synthetic
experiment presented in Fig. 4, it is clear that the use of the USGS data
degrades the performance of the reconstruction. This is likely due to the
nonlinear components of flow, such as reservoirs, dams, or backwater
effects, which are present in this basin and can significantly alter the flow
from what this linear routing model predicts. Additionally, during some of
the peak flow periods (such as days 100 to 150 in Fig. 7b), there are
instances where the reconstructed discharge is greater than the synthetic
truth. This is a result of a numerical correction done in the model where
physically unrealistic negative runoff values resulting from each Kalman
smoother update are reset to a value of zero. The effect of this correction
is more apparent in the assimilation of these USGS observations than in the
synthetic experiments.
Reconstructed discharge time series during 2009 for 2 of the 25
evaluation sites when the ISR model was run using the climatological initial
guess of runoff conditions. Here the discharge data assimilated and compared
against (the truth) are USGS observations.
Figure 8 presents the overall results of these experiments, again displaying
the distributions of KGE values resulting from the initial guess and the
discharge reconstruction. For all initial conditions, the ISR model is able
to create some improvement in the reconstructed discharge values, mainly by
improving upon the correlation and variability; however, the degree of
improvement is noticeably less. In contrast to the synthetic experiments, the
initial guess of runoff derived from the TMPA precipitation resulted in the
best performance for the interpolation while the null and climatological
initial guesses performed similarly, exhibiting a smaller shift in the KGE
values for all the evaluation gauges. In part, this is due to the nonlinear
flow characteristics in the USGS observations that we are not representing,
as there are often conflicting estimates of the spatial distribution of
discharge between the USGS observations and the TMPA precipitation-based
discharge, which lead to a decrease in the innovation term provided by the
Kalman smoother. Another potential cause of this decreased performance could
be errors in the river discharge observations themselves. In situ
observations are likely to have errors of varying magnitudes; however, we
treat these observations as error free for the purposes of model evaluation.
As a result, any potential errors in these observations will then be
transferred to errors in the final reconstructed discharge estimates.
Distributions of NSE and KGE values for the three synthetic
experiments with varied initial conditions of runoff and USGS observations
as the synthetic truth. The ordering is the same as that in Fig. 5.
Discussion
The need for global discharge and runoff observations and estimates is not
new and there have been a number of recent studies that have taken different
approaches to generating spatially and temporally distributed discharge from
point observations (Andreadis et al., 2007; Biancamaria et al., 2011; Paiva
et al., 2013, 2015). The ISR method is an alternative approach to these
methods, which allows for the creation of spatially distributed discharge
fields that are not only spatially consistent but are also consistent through
time, due to the application of a Kalman smoother. The results of these
experiments have shown that the ISR method can produce a representation of
discharge throughout a basin river network given a wide variety of initial
conditions. In particular, the interpolation from USGS observations is
promising, as we are able to generate a very close representation of the
discharge conditions throughout the basin with little to no prior information
about the specific distribution of runoff present. This indicates that the
ISR method may be able to extend the usefulness of observations in basins
with sparse gauge networks, such as in many underdeveloped regions of the
world. It is also important to note that the ISR method produces fields of
runoff that are consistent with the observed discharges, which may prove
beneficial for the calibration and optimization of land surface model
processes in poorly gauged basins.
Although these experiments have illustrated the potential for the ISR method
to be used for river discharge interpolation in global basins, it is
important to acknowledge that these experiments are idealized and thus, do
not contain all of the potential errors and uncertainties that would be
present in a real-world application. As discussed previously, our method is
limited by the lack of a nonlinear routing model, the presence of error-free
observations and the overall parameterization of the routing model (static
parameters for the wave velocity and the diffusivity). While the work of Pan
and Wood (2013) focused on the inversion of spatial runoff fields, the ISR
method is used to generate integrated discharge values; therefore, it is
important to see how sensitive the model will be to similar errors. Figure 9
presents the results of two sensitivity experiments: (1) selection of a
different velocity parameter, and (2) decreasing amounts of available gauge
data. Both of these experiments were carried out using the null initial
conditions of runoff. For the velocity parameter, the calibrated velocity of
1.4 m s-1 produces the best performance; however, there is still skill
in the discharge reconstruction for velocity values from 50 % to
200 % of the optimal value (Fig. 9a). With regards to potentially limited
availability of observations, the number of gauges assimilated in the ISR
model was decreased from 0 % to 50 % of the full set by randomly
removing gauges (Fig. 9b). In all cases, the KGE values indicate adequate
model performance; however, these results are dependent on the information
contained within each observation, as removing a gauge with a larger
contributing area is likely to have a larger impact on the overall model
performance that a smaller one. With regards to errors in the observations,
Pan and Wood (2013) tested the impact of these errors on the ISR
model's ability to reconstruct runoff fields and found that these errors could
potentially be significant enough to remove any positive improvements from
the assimilation procedure. In real-world applications we will need to
carefully consider the error characteristics of the data sources to be
assimilated, as these will have a significant impact on the quality of the
final discharge product.
Box plots of KGE values at 25 validation sites for two sensitivity
experiments: (a) varying wave velocity parameters, (b) removing gauges from
the observation set. In each of these experiments the null initial condition
was used. The mean of each set is denoted with a red X.
Another important variable in the performance of this method is the
availability and selection of gauges (or pseudo gauges in synthetic
experiments) for assimilation and evaluation. In this study we present the
results of assimilating a specific configuration of available gauging sites
as illustrated in Fig. 3. To understand better how the results of these
experiments might change if the network of gauges were configured
differently, we performed a sensitivity study by generating 100 random
configurations of gauges to assimilate and evaluate from the total set of 75.
These gauge networks were then used to reconstruct discharge in each of the
previous six experiments, evaluating the NSE at each of the 25 evaluation
sites, for each possible network configuration. The results of these
simulations are illustrated in Fig. 10, where the gauge configurations for
each experiment are ranked according to the median NSE of reconstructed
discharge. In addition to this, the box and whisker plots illustrate the
performance spread for each configuration, as well as any potential outliers.
Finally, the yellow box in each experiment represents the specific simulation
results that are presented in this paper.
Box
plots of NSE values at 25 validation sites for 100 random configurations of
the gauge network. Experiments are divided into two categories: the entirely
synthetic discharge reconstructions (a, c, e) and the discharge
reconstructions from USGS observations (b, d, f). These experiments
are further differentiated by the three initial runoff conditions used:
null (a, b), climatology (c, d), and TMPA
derived (e, f). The yellow box plots represent the gauge network
configuration used for the results presented in this
study.
Focusing first on the results of the synthetic experiments with a null
initial condition of runoff (Fig. 10a), it is evident that there is a
significant amount of variation in the specific distributions of NSE;
however, there is little change in the median value or the lower limit of NSE
values across all configurations. This indicates that regardless of the
network configuration chosen, we are able to reasonably reconstruct spatially
and temporally distributed discharge within the basin river network. Looking
across the three initial conditions for the synthetic experiments (Fig. 10a,
c, e) shows results comparable to those presented previously, with the null
and climatological initial runoff conditions providing relatively similar
performance. The TMPA-derived initial conditions show a distribution of
median NSEs that is slightly lower than in prior experiments. It is also
interesting to note that with increasing information in the initial
conditions, the model performance spread increases considerably. This is a
further illustration of the case where large differences between the initial
guess and true conditions can degrade the effectiveness of the ISR model in
generating discharge throughout an entire basin.
Finally, the results of these 100 random configurations for the experiments
using in situ USGS discharge observations are shown in Fig. 10b, d, and f.
Overall, there is a pattern of decreasing performance from null to
TMPA-derived initial conditions still present. Across all three experiments,
the range of median NSE values is larger than that for the synthetic
experiments, indicating that the selection of gauges for assimilation in this
real-world scenario has a more significant impact. We also find that the
spread of the NSE distributions is greater than those in the synthetic
experiments, further reinforcing the influence of the previously discussed
error and uncertainty sources in the ISR method. Understanding and
constraining these errors will be critical to future applications of ISR.
Conclusion
In this study we have developed a two-sweep method for reconstructing
spatially and temporally continuous discharge records from discrete
observations of discharge, in which the first sweep applies the ISR method
(Pan and Wood, 2013) to propagate observed information backward in time–space
and the second sweep re-runs the same routing model to propagate information
forward in time–space. The new formulation is expected to offer more complete
propagation of observed information in time and space (thus a better
performance) and a better physical consistency than existing approaches. The
core algorithm of this method is formulated as a Kalman smoother, allowing
for assimilation–interpolation of discharge from all available
observations of discharge in a basin. By assimilating and validating against
synthetic and real observations at 75 gauging sites in the Ohio River basin,
the new approach has illustrated the potential for discharge reconstructions
across all experiments. In particular, the results of the discharge
reconstructions given a null initial runoff condition are promising, as they
illustrate the ability of the streamflow assimilation–interpolation
methodology to create continuous discharge records in a basin where we do not
have an adequate climatology or a calibrated hydrologic model.
The performance of this method will be limited by the availability and
quality of gauge data, the specific initial conditions chosen, the
parameterization of the routing model, and the exclusion of nonlinear
features such as dams (Yin et al., 2016a, b). Further work is needed to
determine how this method will perform as the density of the gauge network is
reduced or as the amount of days missing from a gauge's discharge record is
increased, as would be the case in many of the global basins which do not
currently have robust observation networks. Temporally sparse observations
are particularly challenging for this type of assimilation–interpolation, as
specific extreme events could be missed entirely, or the method may not have
enough data to maintain a correction through time from the initial guess. At
a minimum, ISR can be used to reconstruct a distributed representation of
discharge from one or a few in situ gauge observations; however, the more
information that can be provided for the assimilation, the more likely we are
to produce an accurate estimate of the discharge conditions in that basin.
Ideally, this work should be considered in the context of previous work done
on statistical river kriging, e.g., Paiva et al. (2015) and Yoon et
al. (2013); however, due to various limitations (e.g., lack of properly
trained kriging parameters over the study area), no experiments have been
carried out to compare the performance of ISR-based approach to statistical
river kriging. As such, there are no quantitative metrics to prove the
incremental improvement made by our method.
To improve upon the density of observations in sparsely gauged regions, this
methodology could be extended to perform interpolations from remotely sensed
river discharge products, such as those from current generation satellite
altimetry, or the upcoming NASA Surface Water and Ocean Topography (SWOT)
mission (Alsdorf and Lettenmaier, 2003; Durand et al., 2010; Pavelsky et al.,
2014). The SWOT mission, scheduled to launch in 2021, is of particular
interest, as this will contain a swath altimeter designed to provide global
observations of water surface elevation and slope, from which river discharge
can be estimated. Within the 21 d repeat cycle, a river reach will be
observed 2–4 times, on average (Biancamaria et al., 2010). The prospects for
such a spaceborne sensor are great, especially with respect to global
coverage; however, due to the inclination of the orbit these observations are
not evenly distributed in time or space and thus they will not be as complete
as the USGS observations used here. In general, we believe that this form of
streamflow interpolation using the ISR method could serve as a framework for
creating spatially and temporally continuous discharge records from sparse
observations like the future SWOT mission. Careful consideration will be
required to account for the gaps in observations and the unique error
characteristics of these remotely sensed discharge
observations.
Data availability
The precipitation input datasets are available via the respective websites of the dataset producers. Discharge observations are also available directly through the USGS web interface. Model code and parameter files can be provided through direct request to the authors.
Author contributions
CKF and MP conceived the study. CKF performed the analysis and wrote the paper. MP and EFW commented on the paper and helped with the writing.
Competing interests
The authors declare that they have no conflict of interest.
Financial support
This work was supported by the National Aeronautics and Space Administration (grant nos. NNX16AH84G and NNX15AH05A).
Review statement
This paper was edited by Stacey Archfield and reviewed by Laura Read and one anonymous referee.
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